# Plan I am going to investigate how to make a box with the biggest volume, from a piece of card. The

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Introduction

Plan

I am going to investigate how to make a box with the biggest volume, from a piece of card. The dimensions of the square will range from, 10x 10 20 x 20, 30 x 30 and 40 x 40.

A global formula will then be generated, which will help to calculate where the maximum volume of the open box lies.

First of all, the corners of the square have to be cut off so that it can be folded up to make an open box.

The shaded squares are going to be cut off.

This can be set out as an equation:

Volume of Square = Length x Width x Height

= L x W x H

C= cuts (height)

Length = L-2C

Width = W-2C

Base = (L-2C) (W-2C)

Volume = L x W x H

= (L-2C) (W-2C) C

As L=W, the equation can be simplified to

(L-2C) (L-2C) C

Volume = (L-2C)2

Middle

484

1936

20

20

5

400

2000

18

18

6

324

1944

16

16

7

256

1792

14

14

8

196

1568

12

12

9

144

1296

10

10

10

100

1000

8

8

11

64

704

6

6

12

36

432

4

4

13

16

208

2

2

14

4

56

0

0

15

0

0

This time, the highest figure for volume is when the height, meaning the size of the cut is 5cm.

The maximum value now lies between 5 and 6 cm. so when the square is 30 x 30, the maximum value has increased from between 3 and 4cm to 5 and 6 cm.

Conclusion

To begin with, I can try dividing the length of the square with the size of the cut to see when it reaches a figure that gives the maximum value. This is to be done with all three squares.

20 | 1 | 20 |

20 | 2 | 10 |

20 | 3 | 6.667 |

20 | 4 | 5 |

20 | 5 | 4 |

20 | 6 | 3.333 |

20 | 7 | 2.857 |

20 | 8 | 2.5 |

20 | 9 | 2.222 |

30 | 1 | 30 |

30 | 2 | 15 |

30 | 3 | 10 |

30 | 4 | 7.5 |

30 | 5 | 6 |

30 | 6 | 5 |

30 | 7 | 4.286 |

30 | 8 | 3.75 |

30 | 9 | 3.333 |

40 | 1 | 40 |

40 | 2 | 20 |

40 | 3 | 13.333 |

40 | 4 | 10 |

40 | 5 | 8 |

40 | 6 | 6.667 |

40 | 7 | 5.714 |

40 | 8 | 5 |

40 | 9 | 4.444 |

The highlighted cells show that the finding match my table.

When the length of any sized square is divided by 6, the figure we get is where the maximum value for volume lies.

For example, with the 40 x 40 square, the table of results showed that the maximum volume is between 6 and 7cm, and the trial and error method clearly found it out to be correct.

So in order to find the maximum value for any size square, the equation would be top divide the length by 6

Maximum value = L/6.

I will now experiment with a different ratio of sides. For example, the preivios investigation involved a square with sides of a ratio of 1:1

I will now use 1:2.

A 20 x 30 rectangle.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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